Ralf Hemmecke wrote:
>
> >> So "singleton" category can not be just one element set.
> >
> > ??? By *definition* it is a category that contains just one domain.
> > Are saying that this definition is inconsistent?
>
> Sounds like both of you don't understand each other.
>
> Let me use Aldor here...
>
> ---BEGIN aaa.as
> #include "aldor"
> C: Category == with;
> D: C == add;
> ---END aaa.as
>
> ---BEGIN bbb.as
> C: Category == with;
> D1: C == add;
> D2: C == add;
> ...
> Dn: C == add;
> ---END bbb.as
>
> In aaa.as C is a category that only has exactly one domain in it. But
> this is only true if I consider one fixed program.
> As bbb.as shows that "singleton" category definition depends on the
> context. Thus, Waldek is right, that a category always contains
> (potentially) infinitely many domains (or in other words, one can define
> infinitely many domains to a given category).
>
> Did I understand you intentions correctly?
>
Yes. Let me add another example to make things clearer (using
new syntax for categories with no export):
A : Category == with ()
B : Category == with ()
If program has _only_ the two deffinitions above, then there
are no defined domains which belong to category A and no
defined domains which belong to category B. In other words
sets of defined domains which belong to A is the same as
set of defined domains which belong to B. Naive set
theoretic interpretation would say that A = B, but clearly
the language definition says that A is different than B.
So, something is needed to rescue set theoretic interpretation.
My solution is to consider all domains which may be potentially
defined -- once you add them all nonempty categories are
infinite, but equality of categores is just set theoretic
equality.
--
Waldek Hebisch
[EMAIL PROTECTED]
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